Autocorrelation function of channel matrix in few-mode fibers with strong mode coupling

Qian Hu; William Shieh

doi:10.1364/OE.21.022153

1. Introduction

Few-mode fiber (FMF) is being actively explored as a promising transmission medium to surmount the capacity crunch in future fiber optic communication [1–3]. FMF supports multiple spatial modes co-propagating in one medium simultaneously, enhancing the capacity of information transmission. However, the random mode coupling caused by fiber index imperfections and mechanical perturbations leads to signal dispersion in FMF, called differential modal delay (DMD) [4]. While large DMD and mode coupling can improve the channel diversity, making the system robust against mode-dependent loss (MDL) [5], excessive DMD would increase the computational complexity of digital signal processing (DSP) required for the channel equalization [1]. An in-depth understanding of the statistic properties of FMF channel subject to DMD is essential to assist system design for achieving an optimal system performance.

With respect to single-mode fiber (SMF), there is a large body of studies on the statistics of SMF channel affected by polarization modes dispersion (PMD). The stochastic analysis is conducted in terms of either the PMD vector in the Stokes space [6,7] or the channel matrix in the Jones space [8,9]. Though the Stokes space and the Jones space are isomorphic [10], a PMD vector is more suitable for the analysis of time-domain direct detection systems, whereas a Jones matrix is more useful for the analysis of coherent detection systems. Based on a white Gaussian noise assumption for the birefringence vector with strong coupling, standard tools of stochastic calculus can be used to derive some important statistic properties for the PMD vector and Jones matrix [6–9].

The basic concepts and stochastic study methods for SMF can be extended to FMF. Since each spatial mode supports two orthogonal polarization modes, FMF with N spatial modes supports totally 2N modes with different group velocities. An extended 2N-dimensional vector is used to represent the propagating modes in FMF, and the linear effect of FMF channel can be described as a 2N x 2N channel matrix in the generalized Jones space. Recently, study has been conducted on the statistics of mode dispersion (MD) vector (a general form of PMD vector in SMF) for FMF [11,12]. A generalized Stokes space has been formulated to allow convenient representation of the mode coupling vector (a general form of birefringence vector in SMF) and the MD vector in FMF. Based on a white Gaussian noise model for the mode coupling vector, the statistic properties of MD vector have been derived in [11,12]. The statistics of group delays in FMF has also been derived in [5,13] by dividing fiber into many cascaded sections modeled by random matrices.

On the other hand, the latest decade has seen the revival of coherent detection, where DSP is performed on the channel matrix of fiber optic link [14,15]. Subsequently, the statistical analysis of FMF channel matrix has become critically important. In particular, the statistical analysis provides us the autocorrelation function (ACF) of FMF channel matrix, which gives an insight into the frequency behavior of FMF channel dispersion and helps to find the optimal receiver design. The channel correlation bandwidth derived from the ACF is also important for the coherent detection system design, as it is often needed for the channel estimation, for instance, for the minimum mean-square error (MMSE) based channel estimation [16].

As an extension of the Pauli matrices used as the basis of the Stokes space in SMF [9], high-dimensional trace orthogonal matrices have been proposed in [11] as the basis of the generalized Stokes space in FMF. In this work, we also resort to the trace orthogonal matrices as the basis for FMF channel matrix. In this report, a canonical stochastic differential equation (SDE) for FMF channel matrix is derived in the regime of strong coupling. Applying the standard tools of stochastic calculus to the SDE, we develop an analytical form for the autocorrelation function (ACF) of FMF channel matrix, from which the channel correlation bandwidth is obtained. A comparison of simulation and analytical results demonstrates the accuracy of our theoretical model.

2. FMF channel matrix and decomposition

The main notations used in the paper are as follows: $| V 〉$ denotes a column vector in the generalized Jones space of dimension 2N, $\vec{V}$ denotes a column vector in the generalized Stokes space of dimension $4 N^{2} - 1$ , $\tilde{V}$ denotes a column vector of dimension $4 N^{2}$ , a column vector with elements $v_{i}$ is denoted by $[\dots v_{i - 1}; v_{i}; v_{i + 1} \dots]$ ; Symbol ‘ $†$ ’ represents the transpose conjugate operation for a matrix, and ‘ $Τ$ ’ represents the transpose operation; E[ ] denotes the ensemble average, and Tr{ } denotes the trace of a matrix.

The optical field in an N-mode FMF at length z and frequency $ω$ can be expressed as a 2N-dimensional complex vector $| ψ (z, ω) 〉$ . The linear channel of FMF can be treated as a 2N input 2N output linear system given by [11,12]

| ψ (z, ω) 〉 = U (z, z_{0}, ω) | ψ (z_{0}, ω) 〉,

where U is the channel matrix of dimension 2N x 2N, describing the random mode coupling and group delays in optical channel. Without consideration of MDL, the channel matrix U is a unitary matrix with unit determinant.

Similar to the Pauli matrices used as the basis for the 2x2 Jones matrix of SMF [9], $4 N^{2}$ trace orthogonal matrices form the basis for the 2N x 2N channel matrix of FMF. The trace orthogonal matrices and their associated properties have already been developed in [11]. Specifically, to represent the channel matrix U, $Λ_{0} = I / \sqrt{N}$ and $4 N^{2} - 1$ traceless Hermitian matrices $Λ_{i}$ (i = 1 to $4 N^{2} - 1$ ) are constructed, satisfying the trace-orthogonal condition [11]

T r {Λ_{i} Λ_{j}} = 2 δ_{i j}

where

δ_{i j}

is the Kronecker symbol

(i, j = 0 to 4 N^{2} - 1)

. Note that in (2) we use a normalization factor of 2 instead of 2N in [11]. This follows the convention in other works on special unitary groups [17,18]. Hence, the channel matrix U could be expressed as a superposition of the trace orthogonal matrices [11]

U = \sum_{i = 0}^{4 N^{2} - 1} u_{i} Λ_{i} = \tilde{u} \cdot \tilde{Λ}

where the vector

\tilde{Λ} = [Λ_{0}; Λ_{1}; \dots; Λ_{4 N^{2} - 1}]

is an ensemble of the trace orthogonal matrices, and the complex vector

\tilde{u} = [u_{0}; u_{1}; \dots; u_{4 N^{2} - 1}]

is an ensemble of the weights of each trace orthogonal matrix. Since one set of weights

u_{i}

can exclusively identify a matrix U, the properties of a 2N x 2N channel matrix U is fully represented by a vector

\tilde{u}

of dimension

4 N^{2}

. According to the trace-orthogonal condition (2), the weights

u_{i}

of trace orthogonal matrices

Λ_{i}

could be easily extracted by [11]

u_{i} = \frac{1}{2} T r {Λ_{i} U} .

In group theory, the traceless Hermitian matrices $Λ_{i}$ (i = 1 to $4 N^{2} - 1$ ) form the generators (elements of the Lie algebra) of special unitary group of degree 2N (SU(2N)), satisfying [17]

Λ_{m} Λ_{n} = \frac{δ_{m n}}{N} I + \sum_{k} (i f_{m n k} + d_{m n k}) Λ_{k}

Λ_{n} Λ_{m} = \frac{δ_{m n}}{N} I + \sum_{k} (- i f_{m n k} + d_{m n k}) Λ_{k}

where the indices k, m, n take values from 1 to

4 N^{2} - 1

, and the coefficients f and d are the structure constants. As provided by [11], a cross product “

\times

” of vectors

\vec{A}

and

\vec{B}

in the generalized Stokes space can be defined with the structure constants f as

\vec{A} \times \vec{B} = \sum_{m n k} f_{m n k} A_{m} B_{n} {\vec{e}}_{k} = - \vec{B} \times \vec{A},

where

A_{m}

is the m-th element in vector

\vec{A}

,

B_{n}

is the n-th element in vector

\vec{B}

, and

{\vec{e}}_{k}

is a unit vector with 1 for the k-th element. For the convenience of the mathematical description of our problem, we also introduce a product “

⊙

” of vectors

\vec{A}

and

\vec{B}

defined with structure constants d as

\vec{A} ⊙ \vec{B} = \sum_{m n k} d_{m n k} A_{m} B_{n} {\vec{e}}_{k} = \vec{B} ⊙ \vec{A} .

We call “

⊙

” circle dot product to distinguish it from the conventional dot product ‘

\cdot

’. Then, the matrices

\vec{A} \cdot \vec{Λ}

and

\vec{B} \cdot \vec{Λ}

satisfy the relations

(\vec{A} \cdot \vec{Λ}) (\vec{B} \cdot \vec{Λ}) - (\vec{B} \cdot \vec{Λ}) (\vec{A} \cdot \vec{Λ}) = 2 i (\vec{A} \times \vec{B}) \cdot \vec{Λ},

(\vec{A} \cdot \vec{Λ}) (\vec{B} \cdot \vec{Λ}) + (\vec{B} \cdot \vec{Λ}) (\vec{A} \cdot \vec{Λ}) = \frac{2}{N} (\vec{A} \cdot \vec{B}) I + 2 (\vec{A} ⊙ \vec{B}) \cdot \vec{Λ}

Adding (9) and (10), we get the product of the matrices

\vec{A} \cdot \vec{Λ}

and

\vec{B} \cdot \vec{Λ}

(\vec{A} \cdot \vec{Λ}) (\vec{B} \cdot \vec{Λ}) = \frac{1}{N} (\vec{A} \cdot \vec{B}) I + (\vec{A} ⊙ \vec{B} + i \vec{A} \times \vec{B}) \cdot \vec{Λ}

The choices for the trace-orthogonal matrices are many. The only guide for the choice is the simplicity of calculation. Modified Gell-Mann matrices have been used to build the trace-orthogonal matrices in [11]. In this paper, we select the generalized Gell-Mann matrices, since the structure constants (f and d), extracted from the generalized Gell-Mann matrices, are sufficiently simple for the calculation in appendix B. The generalized Gell-Mann matrices are described in appendix A. The off-diagonal ones are identical to those in [11]. Only the diagonal ones are slightly different from [11]. The generalized Gell-Mann matrices coincide with the Pauli matrices when N = 1.

3. SDE for channel matrix

The evolution of FMF channel matrix U along fiber position $z$ is governed by the mode coupling vector $\vec{β}$ , a generalization of the birefringence vector in SMF

\frac{d U (z, ω)}{d z} = \frac{i}{2} [\vec{β} (z, ω) \cdot \vec{Λ}] U (z, ω)

\vec{β} \cdot \vec{Λ}

is required to be a Hermitian matrix to ensure the unitarity of U. Such that, all the elements in

\vec{β} (z, ω)

are real [11]. For a ‘long’ fiber, where the overall fiber length is significantly larger than the correlation length of local modal dispersion, the white Gaussian noise model is valid for the mode coupling vector given by [11]

\vec{β} (z, ω) = μ ω \vec{n} (z),

where the elements of

\vec{n} (z)

are statistically uncorrelated white Gaussian processes with zero mean and unit variance, and the parameter

μ

represents the mode coupling strength.

Substituting the channel matrix U in (12) with its vector representation (3), and using the product defined in (11), we obtain the evolution equation of the vector representation $\tilde{u} = [u_{0}; \vec{u}]$ as

\begin{array}{l} d u_{0} = \frac{i μ ω}{2 \sqrt{N}} \vec{u} \cdot d \vec{W} \\ d \vec{u} = \frac{i μ ω}{2 \sqrt{N}} [u_{0} d \vec{W} + \sqrt{N} (\vec{u} ⊙ d \vec{W} - i \vec{u} \times d \vec{W})] \end{array}

where

d \vec{W} = \vec{n} d z

is the differential of the Brownian motion

\vec{W} (z)

. Equation (14) can be written in a canonical form of SDE

d \tilde{u} (z) = Q (\tilde{u} (z)) d \vec{W} (z)

where

Q (\tilde{u})

is the diffusion matrix of dimension

4 N^{2}

x

4 N^{2} - 1

with entries to be polynomials of the components in

\tilde{u}

Q = \frac{i μ ω}{2 \sqrt{N}} [\begin{matrix} {\vec{u}}^{Τ} \\ u_{0} I + \sqrt{N} (\vec{u} ⊙ - i \vec{u} \times) \end{matrix}]

We note that, normally a SDE is on a real-value space, whereas the dynamic vector $\tilde{u}$ in Eq. (15) is complex. However, we can consider (15) as a short hand for real-value SDEs. Namely, we can separate the real and imaginary part of the dynamic vector $\tilde{u}$ , and treat the two parts individually as real elements. As long as the Brownian motion term $\vec{W} (z)$ is real, the real and complex forms of SDE describe the same physics of stochastic process. For the sake of concision, we use the complex vector $\tilde{u}$ in our discussion in the remainder of this paper.

Derived from (3), (12) and (13), the m-th column ${\tilde{q}}_{m} (\tilde{u})$ in the diffusion matrix Q satisfies

{\tilde{q}}_{m} \cdot \tilde{Λ} = \frac{i μ ω}{2} Λ_{m} (\tilde{u} \cdot \tilde{Λ}) .

Denote

q_{x, y}

as the entry at x-th row and y-th column in the diffusion matrix Q. Using the trace-orthogonality of

\vec{Λ}

in (17),

q_{x, y}

can be extracted as

q_{x, y} = \frac{1}{2} T r {Λ_{x - 1} ({\tilde{q}}_{y} \cdot \tilde{Λ})} .

Some important statistic properties of the dynamic vector

\tilde{u}

can be derived from the SDE (15) using standard tools of stochastic calculus. However, those standard tools are usually applied for the Ito form of SDE, whereas the product in (15), involving the Brownian motion, should be interpreted as the Stratonovich product [8,9]. Before (15) can be directly used for stochastic calculus, it should be converted from the Stratonovich form into its equivalent Ito form. In the following context, we will use two different approaches to carry out the conversion.

3.1 Stratonovich-Ito conversion algorithm approach

In SDE theory, there is a standard algorithm to convert SDE (15) from the Stratonovich form to its equivalent Ito form [19], which is given by

d \tilde{u} = \tilde{c} (\tilde{u}) d z + Q (\tilde{u}) d \vec{W}

where the diffusion matrix

Q (\tilde{u})

is the same as that in Eq. (15), and

\tilde{c} (\tilde{u})

is the drift correction vector of dimension

4 N^{2}

, whose l-th component

c_{l}

is given by

c_{l} = \frac{1}{2} \sum_{n = 1}^{4 N^{2}} \sum_{m = 1}^{4 N^{2} - 1} q_{n, m} \frac{\partial q_{l, m}}{\partial u_{n - 1}} .

Since

q_{n, m}

can be obtained using (18), the product of the vectors

\tilde{c}

and

\tilde{Λ}

can be written as

\tilde{c} \cdot \tilde{Λ} = \frac{1}{2} \sum_{n = 1}^{4 N^{2}} \sum_{m = 1}^{4 N^{2} - 1} \frac{1}{2} T r {Λ_{n - 1} ({\tilde{q}}_{m} \cdot \tilde{Λ})} \frac{\partial {\tilde{q}}_{m} \cdot \tilde{Λ}}{\partial u_{n - 1}},

where

{\tilde{q}}_{m} \cdot \tilde{Λ}

is already known in (17). Noting

\partial \tilde{u} \cdot \tilde{Λ} / \partial u_{n} = Λ_{n}

, (21) can be rewritten as

\tilde{c} \cdot \tilde{Λ} = - \frac{μ^{2} ω^{2}}{8} [\sum_{m = 1}^{4 N^{2} - 1} \frac{1}{N} u_{m} Λ_{m} + \sum_{n = 1}^{4 N^{2} - 1} \sum_{m = 1}^{4 N^{2} - 1} \frac{1}{2} T r {Λ_{n} Λ_{m} (\tilde{u} \cdot \tilde{Λ})} Λ_{m} Λ_{n}] .

After some length calculation shown in appendix B, we obtain

\tilde{c} \cdot \tilde{Λ} = - \frac{4 N^{2} - 1}{8 N} μ^{2} ω^{2} (\tilde{u} \cdot \tilde{Λ}) .

As such, the drift correction vector

\tilde{c}

in (19) becomes

\tilde{c} = - \frac{4 N^{2} - 1}{8 N} μ^{2} ω^{2} \tilde{u} .

The i-th component in

\tilde{c}

is proportional to the i-th component in

\tilde{u}

.

3.2 Stochastic integral definition approach

We can also develop the correction term $\tilde{c}$ from the first principle of definitions for the Stratonovich and Ito integral.

According to the definition of stochastic integral [20], the integral of canonical SDE (15) can be interpreted as

\int_{a}^{b} Q (\tilde{u} (z)) d \vec{W} (z) \underset{δ \to 0}{= \lim} \sum_{i = 0}^{n - 1} Q (λ \tilde{u} (z_{i + 1}) + (1 - λ) \tilde{u} (z_{i})) Δ {\vec{W}}_{i}

where

z_{i}

(i = 0 to n-1) are n partitions in the intervals [a,b] with mesh

δ = \max (z_{i + 1} - z_{i})

,

Δ {\vec{W}}_{i} = \vec{W} (z_{i + 1}) - \vec{W} (z_{i})

, and

λ

is the parameter defining different methods of integral. The case of

λ

= 0 corresponds to the Ito integral, whereas

λ

= 1/2 corresponds to the Stratonovich integral.

SDE (15) is the canonical form of dynamic Eq. (12) with a vector representation for the channel matrix U. When we substitute the canonical SDE (15) in (25) with (12), the integral of dynamic Eq. (12) is given by

\frac{i μ ω}{2} \int_{a}^{b} [d \vec{W} (z) \cdot \vec{Λ}] U (z) = \frac{i μ ω}{2} \lim_{δ \to 0} \sum_{i = 0}^{n - 1} [Δ {\vec{W}}_{i} \cdot \vec{Λ}] (λ U (z_{i + 1}) + (1 - λ) U (z_{i}))

From (26), the correction term, defined as the conversion between the Ito and Strantonovich integral, is a given by

\frac{i μ ω}{4} [d \vec{W} \cdot \vec{Λ}] d U

Substituting dU in (12) into (27), and using the rules of

d W_{i} d W_{i} = d z

,

d W_{i} d W_{j} = 0

(

i \neq j

) [21, Chapter III], and

\vec{Λ} \cdot \vec{Λ} = (4 N^{2} - 1) I / N

, we can finally reach the correction term as

- \frac{4 N^{2} - 1}{8 N} μ^{2} ω^{2} U \cdot d z

The correction term (28) we get here is consistent with that we developed using the standard Stratonovich-Ito conversion algorithm in last section (24).

It follows that the dynamic equation of U (12) in its Ito form can be expressed as

\frac{d U (z, ω)}{d z} = - \frac{4 N^{2} - 1}{8 N} μ^{2} ω^{2} U + \frac{i μ ω}{2} [\vec{n} (z, ω) \cdot \vec{Λ}] U (z, ω)

The dynamic equation of

\tilde{u}

(14) in its Ito form is

\begin{array}{l} d u_{0} = - \frac{(4 N^{2} - 1) μ^{2} ω^{2}}{8 N} u_{0} d z + \frac{i μ ω}{2 \sqrt{N}} \vec{u} \cdot d \vec{W} \\ d \vec{u} = - \frac{(4 N^{2} - 1) μ^{2} ω^{2}}{8 N} \vec{u} d z + \frac{i μ ω}{2 \sqrt{N}} [u_{0} d \vec{W} + \sqrt{N} (\vec{u} ⊙ d \vec{W} - i \vec{u} \times d \vec{W})] \end{array}

The canonical form of SDE (30) is (19), with the diffusion matrix Q defined by (16), and the drift vector $\tilde{c}$ defined as (24). The standard tools of stochastic calculus can be directly applied to the Ito Eq. (19).

4. Autocorrelation function of channel matrix

The frequency correlation of channel matrix U can be defined as

R (z, ω_{1}, ω_{2}) = U^{†} (z, ω_{1}) U (z, ω_{2})

R is also a unitary matrix, describing the disparity between the channel matrices U at frequencies

ω_{1}

and

ω_{2}

. Its expectation value, E[R], is the ACF of channel matrix, giving a measure of the statistical dependence of U at different frequencies, and indicating how quickly U diverges in frequency domain. The bandwidth of E[R] is called channel correlation bandwidth, representing a frequency span over which U can be approximately considered as a constant matrix from the viewpoint of DSP.

Taking derivative of (31) with respect to length z, the evolution of R along z is

\frac{d R (z, ω_{1}, ω_{2})}{d z} = \frac{d U^{†} (z, ω_{1})}{d z} U (z, ω_{2}) + U^{†} (z, ω_{1}) \frac{d U (z, ω_{2})}{d z}

Substituting the evolution equation of U (12) into (32), we obtain

\frac{d R (z, ω_{1}, ω_{2})}{d z} = \frac{i μ (ω_{2} - ω_{1})}{2} [U^{†} (z, ω_{1}) (\vec{n} (z) \cdot \vec{Λ}) U (z, ω_{1})] R (z, ω_{1}, ω_{2})

Term

U^{†} (z, ω_{1}) (\vec{n} (z) \cdot \vec{Λ}) U (z, ω_{1})

is a rotated white Gaussian noise, and can be denoted as

\vec{n}' (z) \cdot \vec{Λ}

, where

\vec{n}'

can be regarded as the white Gaussian noise vector in a new coordinate system. From (33), we can observe that R is determined only by the frequency difference

Δ ω = ω_{2} - ω_{1}

, such that (33) can be rewritten as

\frac{d R (z, Δ ω)}{d z} = \frac{i μ \cdot Δ ω}{2} [\vec{n}' (z) \cdot \vec{Λ}] R (z, Δ ω)

Equation (34) has the same form as (12), substituting

ω

with

Δ ω

. Following the steps in last section, we can also obtain the Ito equation for (34) in a similar form to (19), given by

d \tilde{r} = \tilde{c} (\tilde{r}) d z + Q (\tilde{r}) d \vec{W}

where

\tilde{r}

is the vector representation of R on the basis matrices

R = \tilde{r} \cdot \tilde{Λ}

. The diffusion matrix

Q (\tilde{r})

is given by

Q (\tilde{r}) = \frac{i μ \cdot Δ ω}{2 \sqrt{N}} [\begin{matrix} {\vec{r}}^{Τ} \\ r_{0} I + \sqrt{N} (\vec{r} ⊙ - i \vec{r} \times) \end{matrix}]

The drift vector

\tilde{c} (\tilde{r})

is given by

\tilde{c} (\tilde{r}) = - \frac{4 N^{2} - 1}{8 N} μ^{2} \cdot Δ ω^{2} \cdot \tilde{r}

The Fokker-Planck equation (FPE) is a powerful tool of stochastic calculus, which can be used to derive the distribution of a dynamic process governed by SDE [21]. Applying FPE to the Ito Eq. (35) gives the probability density function (pdf) p of vector $\tilde{r}$

\frac{\partial p (\tilde{r}, z)}{\partial z} = \frac{1}{2} \sum_{n = 1}^{4 N^{2}} \sum_{m = 1}^{4 N^{2}} \frac{\partial^{2} O_{m n} p (\tilde{r}, z)}{\partial r_{m - 1} \partial r_{n - 1}} - \sum_{k = 1}^{4 N^{2}} \frac{\partial c_{k} p (\tilde{r}, z)}{\partial r_{k - 1}}

where

O = Q Q^{Τ}

. Though the distribution of the vector representation

\tilde{u}

for the channel matrix experiences an instant spread on unit sphere after fiber input due to the white Gaussian noise model adopted for the mode coupling vector [9], the distribution of the vector representation

\tilde{r}

for the ACF shall experience a gradual spread on unit sphere as the random coupling is mitigated by the transpose conjugate operation in (31). The solution to the pdf in FPE (38) will be discussed in a separate future submission.

On the other hand, the Dynkin formula gives the expectation of any smooth function f of the dynamic process $\tilde{r}$ without the knowledge of the pdf [21]

\frac{d E [f (\tilde{r})]}{d z} = E [G f (\tilde{r})]

where G is the Ito generator given by

G = \frac{1}{2} \sum_{n = 1}^{4 N^{2}} \sum_{m = 1}^{4 N^{2}} O_{m n} \frac{\partial^{2}}{\partial r_{m - 1} \partial r_{n - 1}} + \sum_{k = 1}^{4 N^{2}} c_{k} \frac{\partial}{\partial r_{k - 1}}

Since

E [r_{i}]

is what we need to obtain the ACF, the smooth functions in our case should be

f (\tilde{r}) = r_{i}

. The second order differential operator to

r_{i}

in (40) equals to 0, and only the first order differential operator left. Such that, the evolution of

E [r_{i}]

is only determined by the drift term

c_{k}

. Therefore, the Dynkin formula gives

\frac{d E [r_{i} (z)]}{d z} = - \frac{4 N^{2} - 1}{8 N} \cdot μ^{2} \cdot Δ ω^{2} E [r_{i} (z)] .

The correlation matrix R at the input of fiber is an identity matrix I for any frequencies. With the initial condition

r_{0} (0) = \sqrt{N}

and

\vec{r} (0) = \vec{0}

, we obtain

\begin{array}{l} E [r_{0}] = \sqrt{N} \exp (- \frac{4 N^{2} - 1}{8 N} \cdot μ^{2} \cdot Δ ω^{2} \cdot z), \\ E [r_{i}] = 0 (i \neq 0) \end{array}

Consequently, the ACF of channel matrix can be expressed as

E [R] = \exp (- \frac{4 N^{2} - 1}{8 N} \cdot μ^{2} \cdot Δ ω^{2} \cdot z) I

From (43), the 3dB channel correlation bandwidth, where the frequency correlation level falls to 0.5 [22], is obtained

B_{U} = 2 \sqrt{- \frac{8 N \cdot \ln 0.5}{(4 N^{2} - 1) μ^{2} z}} .

Equation (44) shows that the frequency dependence of channel matrix is related to the mode number N, and decreases with the square root of fiber length z.

Taking derivative of (31) with respect to frequencies $ω_{1}$ and $ω_{2}$ , we obtain

\frac{\partial^{2} R}{\partial ω_{1} \partial ω_{2}} = \frac{\partial U^{†} (ω_{1})}{\partial ω_{1}} \cdot \frac{\partial U (ω_{2})}{\partial ω_{2}}

Setting

ω_{1} = ω_{2}

in (45), and noting that the Hermitian MD matrix is defined as

\vec{Ω} \cdot \vec{Λ} = - j 2 U_{ω} U^{†} / \sqrt{N}

where the subscript ‘

ω

’ stands for the derivative over

ω

, we can relate the frequency derivative of R to the MD matrix by

T r {{\frac{\partial^{2} R}{\partial ω_{1} \partial ω_{2}} |}_{ω_{1} = ω_{2}}} = \frac{N}{4} T r {(- j 2 U_{ω} U^{†} / \sqrt{N}) (j 2 U U_{ω}^{†} / \sqrt{N})} = \frac{N \cdot τ^{2}}{2}

where

τ = | \vec{Ω} |

. To derive (47), we have first applied the identity of (46) then of (11). Provided that R is determined only by the frequency difference

Δ ω

,

\partial^{2} R / \partial ω_{1} \partial ω_{2} = - \partial^{2} R / \partial^{2} Δ ω

. Substituting this into (47), we have

E [τ^{2}] = - \frac{2}{N} {T r {\frac{\partial^{2} E [R]}{\partial {(Δ ω)}^{2}}} |}_{△ ω = 0} = \frac{(4 N^{2} - 1) μ^{2} z}{N}

Equation (48) is consistent with the study reported in [11]. The MD vector

\vec{Ω} (z, ω)

of FMF undertakes a

4 N^{2} - 1

dimensional isotropic random walk, and its root-mean-square value

\sqrt{E [τ^{2}]}

grows as the square root of the fiber length z [11,13].

From (43) and (48), the mean-square value of MD vector E[ $τ^{2}$ ] and the ACF of channel matrix E[R] can be related by

T r {E [R]} = 2 N \exp (- \frac{E [τ^{2}] \cdot Δ ω^{2}}{8})

Applying (48) into (44), the correlation bandwidth in relation to the channel matrix can be written as $B_{U} = 4.7 / \sqrt{E [τ^{2}]}$ . On the other hand, the correlation bandwidth in relation to the square modulus of MD vector, as given in [11], is $B_{τ^{2}} = 3.2 \sqrt{(4 N^{2} - 1) / (N^{2} \cdot E [τ^{2}])}$ , which is $0.68 \sqrt{4 - 1 / N^{2}}$ times of $B_{U}$ . We conclude that the MD vector is always slightly more stable in frequency domain compared to the channel matrix. Our conclusions about the ACF of the FMF (43) and (49) coincide with those of the SMF studied in [9] when N = 1. We extend the work of [9] to a more general case of arbitrary number of modes. We note that the MD vector $\vec{Ω}$ defined in (46) has a factor of $\sqrt{N}$ difference than that in [11]. With the MD vector definition in (46), the ACF is independent of the number of modes for the same mean-square value of MD vector E[ $τ^{2}$ ]. Namely, E[ $τ^{2}$ ] will be a good indicator of the correlation bandwidth regardless of the number of modes under the condition of strong coupling.

5. Simulation

We conduct the Monte-Carlo simulation to verify our theoretical result obtained in the last section. A FMF is divided into 100 sections, assuming that the length of one section is longer than the correlation length. The local principle modes (PMs) in separate sections are considered independent and uniformly distributed. The propagation of each section is modeled as a 2N x 2N matrix with equivalent statistical properties, given by [5,13]

V (ω) = C_{1} T (ω) C_{2}^{†}

where

C_{1}

and

C_{2}

are independent random unitary matrices corresponding to the random coupling of the PMs at the input and the output of the section, and T is a diagonal matrix representing the impact of group delays on the PMs

T (ω) = d i a g [e^{- j ω t_{1}}, e^{- j ω t_{2}}, ..., e^{- j ω t_{2 N}}]

where

t_{i}

(i = 1 to 2N) are the group delays of the local PMs. The average delay is a constant by setting

\sum t_{i} = 0

. The channel matrix of the whole fiber is obtained by multiplexing the propagation matrices V for each section. The choice of the individual group delays does not have significant impact on the statistical properties of the global fiber as long as the modes are strongly coupled [13]. We just need to ensure that the MD in each section

T = \sqrt{2 \sum t_{i}^{2} / N}

satisfies the requirement of the global MD

τ = T \sqrt{100}

.

After the channel matrices U at different frequencies are obtained using the Monte-Carlo simulation, their correlation R could be calculated using (31). We measured the ACF for FMFs with a 10-ps global MD in the bandwidth of 250 GHz. 10000 fiber configurations are used in the simulation to obtain the ensemble averaging. The simulation results and theoretical predictions of $T r {E [R]} / 2 N$ for FMFs with 2 to 4 spatial modes are plotted in Fig. 1. The accuracy of our theory result is justified by the excellent match between the simulation and theoretical results.

Fig. 1 The comparison of the analytical and simulation result for the ACF of the FMF channel matrix with 2 to 4 spatial modes.

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6. Conclusion

In this paper, we have used two different approaches to derive the SDE for the FMF channel matrix in the regime of strong coupling. The channel matrix is decomposed over the generalized high-dimensional Gell-Mann matrices, an equivalent of two-dimensional Pauli matrices. We also develop the ACF of channel matrix for a strong coupling FMF. The channel correlation bandwidth obtained from the ACF indicates that the frequency dependence of the channel matrix decreases with the square root of fiber length. The validity of our analytical result is confirmed by the Monte-Carlo simulation.

Appendix A

We choose the generalized Gell-Mann matrices as the basis matrices for the decomposition of the channel matrix. The n-dimensional Gell-Mann matrices are constructed by the following algorithm:

The first n-1matrices are diagonal with the form

Λ_{k} = \sqrt{\frac{2}{k (k + 1)}} [\begin{matrix} 1 \\ ⋱ \\ 1 \\ - k \\ 0 \\ ⋱ \\ 0 \end{matrix}] (k = 1 to n - 1) .

where the first k diagonal entries are 1, and the k + 1th diagonal entry is –k. The rest entries are all 0. The coefficient

\sqrt{2 / (k (k + 1))}

is added to satisfy the trace-orthogonal condition

T r {Λ_{i} Λ_{j}} = 2 δ_{i j}

.

The rest $n^{2} - n$ Gell-Mann matrices are off-diagonal. Half of them are symmetric

Λ_{p q} = [\begin{matrix} 1 \\ 1 \end{matrix}],

where the entries at the p-th row, q-th column and the q-th row, p-th column are 1, and the rest entries are 0. The other half are antisymmetric

Λ_{p q} = [\begin{matrix} - i \\ i \end{matrix}],

where the entry at the p-th row and q-th column is –i, the entry at the q-th row and p-th column is i, and the rest entries are 0.

Following this construction algorithm, we can easily prove out that $\sum_{m} Λ_{m} Λ_{m} = 2 (n^{2} - 1) I / n$ .

Appendix B

The product $Λ_{n} Λ_{m}$ and $Λ_{m} Λ_{n}$ in (22) can also be expanded as (5) and (6). Using the trace-orthogonality of $\vec{Λ}$ , the structure constants $f_{m n k}$ and $d_{m n k}$ can be extracted by

d_{m n k} = \frac{1}{4} T r {Λ_{k} (Λ_{m} Λ_{n} + Λ_{n} Λ_{m})}

f_{m n k} = - \frac{i}{4} T r {Λ_{k} (Λ_{m} Λ_{n} - Λ_{n} Λ_{m})} .

Switching indices m with n in the above functions, we have

d_{m n k} = d_{n m k}

, and

f_{m n k} = - f_{n m k}

. Substituting n with m in (53), we have

f_{m m k} = 0

. Then,

Λ_{m} Λ_{m} = I / N + \sum_{k} d_{m m k} Λ_{k}

. Since we know

\sum_{m} Λ_{m} Λ_{m} = (4 N^{2} - 1) I / N

, we can easily prove out

\sum_{m} d_{m m k} = 0

.

Substituting (5) and (6) into the second term on the right side of (22), we have

\begin{matrix} \sum_{n = 1}^{4 N^{2} - 1} \sum_{m = 1}^{4 N^{2} - 1} \frac{1}{2} T r {Λ_{n} Λ_{m} (\tilde{u} \cdot \tilde{Λ})} Λ_{m} Λ_{n} = \sum_{n = 1}^{4 N^{2} - 1} \sum_{m = 1}^{4 N^{2} - 1} [\frac{δ_{m n}}{\sqrt{N}} u_{0} + \sum_{k = 1}^{4 N^{2} - 1} (- i f_{m n k} + d_{m n k}) u_{k}] \\ \cdot [\frac{δ_{m n}}{\sqrt{N}} Λ_{0} + \sum_{l = 1}^{4 N^{2} - 1} (i f_{m n l} + d_{m n l}) Λ_{l}] . \end{matrix}

Using the properties of coefficients

f_{m n k}

and

d_{m n k}

we have discussed above, (54) can be simplified as

\sum_{n = 1}^{4 N^{2} - 1} \sum_{m = 1}^{4 N^{2} - 1} \frac{1}{2} T r {Λ_{n} Λ_{m} (\tilde{u} \cdot \tilde{Λ})} Λ_{m} Λ_{n} = \frac{(4 N^{2} - 1)}{N} u_{0} Λ_{0} + \sum_{k l} \sum_{m n} (f_{m n k} f_{m n l} + d_{m n k} d_{m n l}) u_{l} Λ_{k} .

For the Gell-Mann matrices described in appendix A, we can easily prove out that

\sum_{m n} (f_{m n k} f_{m n l} + d_{m n k} d_{m n l}) = 0

for the case of

k \neq l

, and

\sum_{m n} (f_{m n k}^{2} + d_{m n k}^{2}) = (4 N^{2} - 2) / N

for the case of

k = l

. Thus, (22) can be written as

\tilde{c} \cdot \tilde{Λ} = - \frac{4 N^{2} - 1}{8 N} μ^{2} ω^{2} [u_{0} Λ_{0} + \sum_{k = 1}^{4 N^{2} - 1} u_{k} Λ_{k}] .

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Autocorrelation function of channel matrix in few-mode fibers with strong mode coupling

Abstract

1. Introduction

2. FMF channel matrix and decomposition

3. SDE for channel matrix

3.1 Stratonovich-Ito conversion algorithm approach

3.2 Stochastic integral definition approach

4. Autocorrelation function of channel matrix

5. Simulation

6. Conclusion

Appendix A

Appendix B

References and links

Cited By

Figures (1)

Equations (59)

Optics Express